Not a few numerical methods have been developed to empower modeler in solving complex mathematical expressions and in conducting analysis. Nevertheless, even more complex system conditions, fluid behavior and mathematical models may outperform the goodness of any given methodology. This is the case of the so called Non-Newtonian fluids under the influence of buoyancy driven forces. Motivated by the need of understanding the hydrodynamic developed by non- Newtonian fluids in non-isothermal conditions, this contribution focuses on the application of the Taylor's expansion series as a strategic solution of the "Power Law hydrodynamic model" under free convective forces for different types of non-Newtonian fluids. The implication of this contribution can be linked to the search of new chemical products, the exploration of their physicochemical properties, and the design of the most basic unit operation. Furthermore, the study concentrates on the comparison of the numerical solution and the Taylor's solution to validate the existence of multiple solutions as predicted by this last method. Several cases of Joule heating generation are examined and its influence on the velocity profile. Especial attention is given to fact that traditional numerical methods do not facilitate the reconciliation of the material balance under batch conditions. Illustrative results involving different geometries are presented to demonstrate the validation of the proposed approach when compared with typical numerical methods.